Set Mathematics
Set mathematics was developed and refined in the second half of the nineteenth century by mathematicians and philosophers such as Georg
Cantor, Guiseppe Peano and Bertrand Russell. These researchers were interested in establishing a rigorous logical foundation for mathematics,
and in demystifying certain previously confusing mathematical concepts such as infinity and continuity. They came to realize that poorly understood
mathematical ideas would be clarified if they could be expressed in terms of certain simpler, universal, unambiguous terms and symbols,
and they also realized that the concepts from set mathematics provided these basic building blocks from which all of mathematics
could be rigorously established.
Georg Cantor in particular can be considered the father of set mathematics. Although his work was not widely appreciated in
his own time (except by a few other visionaries such as Russell and Peano), it is clear today that his discoveries were fundamental to the
mathematical progress that was achieved in the twentieth century.
In particular, Cantor showed that the concept of infinity, which had confused mathematicians and philosphers for centuries, could
be easily understood and studied if it was defined in terms of set mathematics. By defining an infinite set to be any set that has the
same cardinality as one of its proper subsets, he was able to develop an impressive theory of infinite numbers, which included the
discovery that in fact there are many different levels of infinity.
Bertrand Russell offers this concise evaluation of Cantor's study of infinite numbers:
George Cantor defined an 'infinite' collection as one which has parts containing as many terms as the whole collection contains.
On this basis he was able to build up a most interesting mathematical theory of infinite numbers, thereby
taking into the realm of exact logic a whole region formerly given over to mysticism and confusion.
On a more mundane level, Cantor established the convention of using set braces ("{" and "}") to contain the elements of sets. The symbols
that we use to represent union, intersection and set membership were introduced and refined by Peano and Russell.
For further reading:
Georg Cantor
Guiseppe Peano
Bertrand Russell